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台灣大學 · 地理環境資源學系 · 轉學考考古題 · 民國114年（2025年）

114 年度 微積分(C)

台灣大學 · 地理環境資源學系 · 轉學考

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Part I

5 points for each blank.
第 1 題5 分
Evaluate. lim⁡x→1x+73−21−x=\lim\limits_{x \to 1} \frac{\sqrt[3]{x+7}-2}{1-\sqrt{x}} = (1) , lim⁡x→∞xtan⁡−1xπ=\lim\limits_{x \to \infty} x\tan^{-1} \frac{x}{\pi} = (2) .
第 2 題5 分
The absolute maximum value of f(x)=sin⁡x2−cos⁡xf(x) = \frac{\sin x}{2 - \cos x} over the closed interval [0,π][0, \pi] is (3) . The absolute minimum value of f(x)f(x) over [0,π][0, \pi] is (4) .
第 3 題5 分
Let RR be the region bounded by y=ln⁡xy = \ln x, y=0y = 0, and x=e2x = e^2. The volume of the solid obtained by rotating RR about the xx-axis is (5) . The volume of the solid obtained by rotating RR about the yy-axis is (6) .
第 4 題5 分
Evaluate. ∫02∫x2x2∫0sin⁡(y2)2025 dz dy dx=\int_0^2 \int_{x^2}^{x^2} \int_0^{\sin(y^2)} 2025 \, dz \, dy \, dx = (7) , ∫−11∫∣y∣12−y2(x2+y2)3/2dx dy=\int_{-1}^1 \int_{|y|}^1 \frac{\sqrt{2-y^2}}{(x^2 + y^2)^{3/2}} dx \, dy = (8) .

Part II

15 points for each problem.
第 5 題15 分
Given that the Maclaurin series of sin⁡x\sin x is ∑n=0∞(−1)nx2n+1(2n+1)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}.
(a)
Explain why this fact leads to sin⁡xx≈1−x26+x4120\frac{\sin x}{x} \approx 1 - \frac{x^2}{6} + \frac{x^4}{120} for small positive xx-values.
(b)
Sketch the graph of f(x)=1−x26+x4120f(x) = 1 - \frac{x^2}{6} + \frac{x^4}{120}. Label the local extrema and inflection points.
第 6 題15 分
Consider the differential equation dydx=y−2x\frac{dy}{dx} = y - 2x.
(a)
Suppose that f(x)=mx+bf(x) = mx + b is a solution to the differential equation. Find the values of mm and bb.
(b)
Find the most general solution using the formula for linear differential equations: y′+P(x)⋅y=Q(x)⇒y=1I(x)∫I(x)⋅Q(x) dxy' + P(x) \cdot y = Q(x) \Rightarrow y = \frac{1}{I(x)} \int I(x) \cdot Q(x) \, dx, where I(x)=exp⁡(∫P(x) dx)=e∫P(x)dxI(x) = \exp \left( \int P(x) \, dx \right) = e^{\int P(x) dx}.
第 7 題15 分
Consider the part of the surface x2y2=2x^2 y^2 = 2 in the first octant (x>0,y>0,z>0x > 0, y > 0, z > 0).
(a)
Use Lagrange multipliers to find the point on the surface x2y2=2x^2 y^2 = 2 that is closest to the origin.
(b)
Let f(x,y)=x2+y2+2x2yf(x, y) = x^2 + y^2 + \frac{2}{x^2 y}. Find all critical points of ff (points with fx=fy=0f_x = f_y = 0). How is this related to part (a)?
(c)
Find fxx,fxy,fyyf_{xx}, f_{xy}, f_{yy} and D=fxxfyy−[fxy]2D = f_{xx}f_{yy} - [f_{xy}]^2 at the critical point.
第 8 題15 分
Find the average value of the function F(x,y,z)=x+y+zF(x, y, z) = x + y + z over the solid E:x2+y2+z2≤1,x≥0,y≥0,z≥0E: x^2 + y^2 + z^2 \leq 1, x \geq 0, y \geq 0, z \geq 0. The formula for average value is 1Volume(E)∭EF(x,y,z) dV\frac{1}{\text{Volume}(E)} \iiint_E F(x, y, z) \, dV.
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